Table Of ContentTHE WORLD-TIME PARALLEL
A.A.Rini and M.J.Cresswell
CONTENTS
Preface ii
Introduction 1
Part I: Truth and Indexicality
Chapter 1 Semantical indices 12
Chapter 2 Philosophical entities 27
Chapter 3 Situated truth 39
Chapter 4 The privileged position 49
Part II: Predicate Logic: Tense and Modal
Chapter 5 A formal language 59
Chapter 6 The non-existent 70
Chapter 7 Multiple indexing 84
Chapter 8 Time and world quantifiers 94
Part III: Times and Worlds, or Tense and Modality?
Chapter 9 Primitive modality and primitive tense 107
Chapter 10 ‘Modalism’ and ‘Tensism’ 121
Chapter 11 The present and the actual 130
Chapter 12 Utterances 142
Chapter 13 Relativity 152
Part IV: De Rerum Natura
Chapter 14 Individuals and stages 163
Chapter 15 Predicate wormism 174
Chapter 16 Abstract and concrete 185
Chapter 17 Supervenience 196
Appendices
Appendix 1 The equivalence of �multi, �xtw and �i 209
Appendix 2 Language and metalanguage 224
Appendix 3 Plantinga’s metaphysics 235
Appendix 4 Interval semantics 238
Appendix 5 Fatalism and the world-time
parallel (with H. Kocurek) 243
Bibliography 252
Index 261
Preface
The emergence of possible-worlds semantics for modal logic in the late 1950s
and early 1960s led to a recognition that the structure of modal and temporal
logic can be treated in an exactly parallel way, with ‘possible worlds’ in the one
case playing the same role as moments of time in the other. While this has been
known now for many years there has been considerable reluctance among
philosophers to ask why it should be so, and to embrace its consequences. Most
of those who have written on the topic have had the aim of attempting to explain
why the formal parallel has little philosophical significance. The present volume
is, we believe, the first book-length work to address the phenomenon explicitly
and present the case for its power.
The work has been supported by a grant from the New Zealand
Government’s Marsden Fund, administered by the Royal Society of New
Zealand. We would express our thanks to the Marsden Fund, and to our three
universities, Massey University, the University of Auckland and the Victoria
University of Wellington for providing the resources for the research to be
undertaken. Some of the material was also used in a graduate course at Texas
A&M University in the (northern) spring semester of 2007, and we are grateful
to A&M for that opportunity.
ii
Introduction
Bird flu is feared, not for what it is, but for what it might become.
2006 news headline
Today is sunny at Waitarere Beach. The sea is sparkling, the sky is blue, the air
is calm and it feels good to be alive.
(1) But it wasn’t sunny yesterday. Yesterday was grey, wet and
windy, and we were depressed.
(2) But it didn’t have to be sunny today. It might have been grey, wet
and windy, and we would have been depressed.
What have (1) and (2) in common? Begin with (1). Suppose that today is
Tuesday, meaning obviously some particular Tuesday. Then (1) is true on
Tuesday because there was rain (and it was grey and windy and so on ...) on
Monday. That seems like common sense. It also seems common sense that rain
is something that can occur on one day — Monday — and fail to occur on
another day — Tuesday. What then should be said about (2)? We shall assume
in this book that sentences about what is necessary or about what is possible have
1
literal truth values. Our aim is to introduce you to a way of dealing with (2)
which is exactly parallel with the way of dealing with (1) that we have just
mentioned. This came about with the advent of the possible-worlds semantics for
modal logic — the logic of necessity and possibility — in the early 1960s.
Corresponding to the times Monday and Tuesday and so on, at which things
happen, philosophers began to speak of the possible worlds in which they happen
— so that (2) is true because in some possible but non-actual world there is rain
today (at Waitarere), even though in the actual world it is sunny. Where is this
world? Well, where is yesterday’s rain? Now it is nowhere, but yesterday it was
here in Waitarere. Similarly the merely possible rain would have been here.
Because times and worlds are often in the domain of metaphysicians,
metaphysical intuitions are aroused that make philosophers suspicious of the
1
Chapter 3 of Lycan 1994 is a sustained demonstration of the ubiquity of modal discourse —
of sentences about what is or is not possible or necessary. Not all philosophers might agree on the
importance of modality. Some feel that although modality might be a part of natural language it
should be banished from austere sc ientific discourse. Smart 1987, p. 182, says “... I want to keep
modality (and possible worlds) out of physical theory...”. And some linguists have suggested that
a sentence like (2) may not be true or false — but may be used merely to express an attitude.
1
INTRODUCTION
world-time parallel, or unsure of its real import. For a start, many philosophers
don’t like the assumption of other possible worlds. One of the earliest was Arthur
Prior, who, in Prior 1957, had already advocated a temporal interpretation of the
operators of modal logic. Prior saw that you could think of modality in the way
in which it is common to think of tense. In Prior 1968c he comments on p. 191:
I wonder whether anybody wants to put forward anything like the
following as a piece of serious metaphysics: There really are such objects
as possible worlds, and what we loosely describe as propositions of
modal logic are in fact predicates of which these objects are the subjects.
He follows this up on p. 192 with
... this seems a tall story, and as I have said, I doubt whether anyone
seriously believes it. But plenty of people believe an exactly similar story
about tenses, i.e. believe that tensed propositions are predicates of
`instants', and that there is — really is — an instant at which I unalterably
`am' drinking.
Only a very few years after Prior made these observations David Lewis explicitly
referred to this passage, and proclaimed that he was one who did accept that other
2
worlds were equally as real as other times. Prior and Lewis represent extremes
in this debate: Prior thinking that neither other times nor other worlds are ‘real’;
and Lewis thinking that both are real. What they share is a belief that the cases
are parallel, and it is the parallel itself that the present book will be investigating.
But surely it might be said that it is obvious that worlds and times are
different — that it is obvious that modal and temporal talk have a different
ontological status. Consider the execution of Charles I in 1649. This actually
happened. It is part of reality. Charles I was executed. Some years later Oliver
Cromwell was offered the crown. He refused. Cromwell might have become
king, but he didn’t. King Oliver is not part of reality. Doesn’t this show that the
whole enterprise of examining the world-time parallel is flawed from the
beginning?
Our first comment is this. It is no part of the claim that worlds and times
are structurally parallel that temporal talk is synonymous with modal talk. Worlds
2
In footnote 6 on p. 185 of Lewis 1970, Lewis says, “A.N. Prior states the indexical theory of
actuality in [Prior 1968c] but, sadly, he goes on to say ‘this seems a tall story, and ... I doubt
whether anyone seriously believes it.’”
2
INTRODUCTION
and times are different. Tense is about times, and modality is about worlds. But
tense and modality have parallel structures — that is, every temporal locution has
a corresponding modal locution, and vice versa.
(3) Charles I was executed
does make a claim about what actually happened, while
(4) Oliver Cromwell might have become king
does not. Put in terms of times and worlds (3) is now true in the actual world
because Charles I was executed (in the actual world) at some time t which
3
precedes the present. (4) is true because there is a possible but non-actual world
in which Cromwell accepted the crown. (3) has a structure
(5) It was so that "
while (4) has a structure
(6) It might have been so that ".
The examples we have chosen here draw out the parallel between tense and
modality by setting out a past tense sentence and a sentence about an unrealized
possibility. The parallel between tense and modality is itself more general so that,
instead of the structure of (5) and (6) we can describe the structure of (3) and (4)
by exhibiting more general schemata:
(7) it is/was/will be that "
3
It is customary in (at least philosophical) English to speak of truth ‘at’ a time, and truth ‘in’ a
world. As far as this book is concerned ‘true at’ and ‘true in’ are no more than stylistic variants.
Some philosophers have tried to make more of this distinction. For instance Adams 1981, pp. 20-
24, restricts truth in a world to truths about things which exist in the world in question. This is
because he thinks that propositions about things which do not exist in a particular world do not
themselves exist in that world, and so cannot be true or false in that world. To the best of our
knowledge no philosopher has made a distinction between existing at a time and existing in a time,
but the analogous distinction could certainly be made. Prior for one (Prior 1957, p. 31) claims that
there are no facts about things which no longer exist. Prior himself took this to demand a logic in
which bivalence has been given up, and he began attempting to study such a logic in Chapter 5 of
Prior 1957. W e are assuming a classical bivalent attitude to truth, and our use of ‘true in a world’
is Adams’s ‘true at a world’.
3
INTRODUCTION
(8) it is/might have been that ".
(7) invokes the tense trichotomy of past, present and future, while (8) invokes the
modal dichotomy of actual and possible. On the face of it this might seem to
make the project implausible, since it is lopsided. Explaining how this dichotomy
and trichotomy are nevertheless parallel will be one of the tasks of this book. So
let us go back to the original claim that (3) is a part of reality but that (4) is not.
What might this mean? Well certainly
(9) It is true in reality that Charles I was executed.
But then
(10) It is true in reality that Cromwell might have become king
is equally so, as is
(11) Cromwell might have really become king.
What is not true is
(12) Cromwell became king
but all that that tells us is that something might have been true which never was
true. That certainly is a truism, but it merely reflects the fact that worlds and
times are different, and is of little help in assessing their comparative reality.
Have we been too swift? Charles I’s execution is actual, Cromwell’s
becoming king is merely possible. Isn’t that enough? Perhaps, but we here
introduce the other way of marking the difference. In modality we talk of the
actual and the possible. Sometimes we speak of the merely possible, as what is
possible but not actual. The temporal contrast is between the present and the past
and future. The first point is that talking of past, present and future cannot
provide an analysis of tense. Although the execution of Charles I is not present
— it is past — yet it was present. In fact anything that is past was once present.
There is a famous argument due to J.M.E. McTaggart against the reality of time.
One premise of this argument is that everything must be both past and present
(and future as well), while another premise is that nothing can be both past and
present. Whatever this curious argument does or does not show about the
unreality of time, the first premise is only plausible in a tensed language, where
4
INTRODUCTION
4
it can be said that everything that is past was present.
What does all this tell us about modality? It is this. The claim is supposed
to be that the actual encompasses everything that is so — that there is nothing
apart from what is actual. But that turns on what is means. Although only the
actual perhaps is actual, there is much that is not actual but might have been
actual. Cromwell might actually have become king. Although he didn’t, it might
have been actual that he did. One might respond that this is to conflate what is
actual with what might have been actual. And this helps illustrate how the
parallel plays out. For consider the claim that Charles I’s execution is present.
That claim is false. What is true is, presumably:
(13) Charles I’s execution was present
and (13) seems to mean the same as
(14) Charles I’s execution is past.
As was said in the discussion of (7) and (8), the modal phrase corresponding to
‘was’ is ‘might have been’. In (13) the ‘presentness’ of Charles I’s execution is
qualified by ‘was’, and, for the same reason, in
(15) Cromwell’s becoming king might have been actual
the ‘actuality’ of Cromwell’s becoming king must be qualified by ‘might have
been’.
But perhaps there is this difference. Perhaps there is no such thing as
Cromwell’s becoming king, and so (15) is false, or maybe does not even make
a claim, since there is nothing which might have been actual except what is
actual. Someone who says this will be claiming that while (14) is an alternative
way of expressing (3), yet (15) is not an alternative way of expressing (4). The
reason is that Cromwell’s becoming king only exists if Cromwell does become
king, and since he never did there is nothing which might have been actual. If this
4
Chapter 33 of McTaggart 1927 argues that the alleged contradiction is essential to time itself,
and therefore that time is unreal. Certainly no predicate like ‘present’ can analyse tense. So if tense
needs an analysis we are in for a vicious regress, just as Mellor 1981 claims on p. 94. As an
analysis of tense the regress is vicious because, as Bigelow 1991, p. 5, says, it “presupposes what
we are trying to explain.” This is in fact McTaggart’s own complaint at p. 21. Dyke 2002a argues
(p. 141) that even if you can avoid the contradiction at the first level it is only by introducing one
at the second level, and so on. She is right that if there is a contradiction at any level it won’t be
removed by going a level up.
5
INTRODUCTION
is the argument look again at (14). For there is no such thing as Charles I’s being
executed — that is Charles I is not being executed. Of course there was such a
thing, in that Charles I was executed. The defender of (14) can say that the talk
of Charles I’s execution does not entail that it is now taking place. All that it
entails is that it did take place, or is taking place, or will take place. In fact it did
take place. That is what the truth of (3) amounts to. But that defence is available
to the defender of (15). For the claim is not that Cromwell’s becoming king is
taking place, or that it was, or that it will be. It is merely that it might have taken
place. That is what the truth of (4) amounts to. And nothing has been said so far
to show what is wrong with that.
What do we mean by a “structural parallel”? It might help to begin with
some explanation of what the parallel is not. Since we began writing this book
we have often received comments like the following:
So you are going to argue in favour of the world-time parallel. That’s
nice, but it’s hardly new. Most metaphysicians already recognise that
there is a structural parallel at the level of logic. But we claim that you
can admit this parallel without supposing it tells you anything of
metaphysical significance. Yes there is a logical parallel between what
happens at other times and what happens in other worlds, but there is all
the metaphysical difference in the world between the reality of these
other worlds and the reality of other times. So we can accept your parallel
and yet feel it doesn’t help us at all, and that we need take no notice of it.
Some people suppose that when we say there is a structural parallel we mean
simply that tense and modality can be formally modelled using the resources of
semantical indices – i.e., using possible worlds for modality, and using times for
tense. Others may want to know what is “the correct” tense logic or “the correct”
modal logic. And they suppose that a claim that there is a logical parallel between
time and modality must be wrong when they make the further supposition that
a system like S5 might be “right” for modality, but is surely “wrong” for time
with both a past and a future operator. (See footnote 11 on page 21 for more
about this.) Our book is not about logic in this sense. What we mean is this. If
you are faced with an argument in the philosophy of modality, there ought to be
a corresponding argument in the philosophy of time which has the same
structure.
Suppose you have two metaphysical arguments, one, T say, about time,
and another parallel argument, M say, about modality. The metaphysicians we
have in mind agree that the logical parallel is exact. So imagine a step T in
n
6
INTRODUCTION
argument T. Then there will be a corresponding step M in argument M, and vice
n
versa. Either T — and therefore M — is a premise, or it is obtained from some
n n
earlier members of T. If the latter, then, given that the logical parallel is exact,
M will be obtainable from the earlier members of M by the same logical rules
n
according to which T was obtained from the earlier members of T. Either these
n
rules are good rules, or they are not, but either way the success of the modal
argument stands or falls with the success of the temporal argument. So T must
n
be a premise, and further, it must be a premise for which no argument is given,
since if an argument is given you can apply the same procedure to that argument.
So it must be an unargued premise. Of course it is easy to provide such. If you
postulate as unargued premises:
T : All times are ontologically real
0
and
M : Only the actual world is ontologically real
0
then indeed you will lose the parallel. But unfortunately you then have nothing
better to offer the advocate of a different premise than an incredulous stare, and
your opponent will simply stare right back at you. To put the point crudely we
suggest that if logic is about argument, a metaphysician who gives the T /M
0 0
reply is suggesting that logical argument has no place in metaphysics.
We do not in this book claim that there are never reasons to treat temporal
and modal arguments differently. But we do claim that whenever you meet one
of these arguments you should very carefully examine the corresponding one.
More importantly, as we try to illustrate, it is by no means a trivial matter to be
clear about just what the corresponding argument is.
Part I of the book sets out what we call an ‘indexical’ semantics. That is
a semantics in which a sentence is held to be true or false, not absolutely, but at
a semantical index, or more accurately at a sequence of semantical indices. Of
these we single out three, a time, a possible world, and a ‘person’, where a person
may be understood abstractly, as for instance in the sense of ‘observer’ used in
relativity theory. It is important to say something here about our use of the word
‘indexical’. While we believe that it is the use envisaged by early writers in the
5
semantics of intensional logic , it has acquired a number of different uses in
recent years. For instance MacFarlane 2009, p. 252 suggests the following
definition:
5
We make some historical remarks about indexicality on p. 34 below.
7
INTRODUCTION
(1) An expression is indexical iff its content at a
context depends on features of that context.
(2) An expression is context-sensitive iff its
extension depends on features of that context.
We do not, in this book, use ‘content’ as a technical term, but in Chapter 2 we do
offer an account of various kinds of propositions as sets of indices — an untensed
proposition is a set of worlds, a tensed proposition is a set of world-time pairs,
and so on. An indexical semantics assigns such entities as the semantic content
of sentences, and while the truth or falsity of such sentences changes from index
to index, the proposition expressed by the sentence does not — since the sentence
is true or false at an index iff that index is in the set assigned to the sentence. This
means that we are using the adjective ‘indexical’ to mean what MacFarlane
6
means by ‘content-sensitive’. Whether there is, in addition to this kind of
indexicality, a need for indexicality in MacFarlane’s sense is a question we do
not deal with in this book, though we make a few remarks about the content of
utterances in Chapter 12.
We end Part I by looking at the possibility that truth at an index can be
given an analysis in terms of a primitive notion of actual present truth. This raises
the following question: Do we privilege the actual in a way in which we do not
privilege the present? Critics of possible-worlds metaphysics may say that it
cannot analyse ‘plain truth’ but only truth in a world. Some philosophers of time
– the so-called presentists, such as Craig (2000a and 2000b) – say that others –
the so-called eternalists, such as Mellor (1981, 1998) – cannot analyse ‘present
truth’ but only truth at a time. The upshot of our discussion suggests that it may
not matter whether you take truth at an index as basic, or simple actual present
truth. But more importantly it suggests that whatever you do about this you can
do equally for tense and for modality.
To give precision to our discussion of the world-time parallel we give in
Part II a formal definition of the syntax and semantics of first-order predicate
languages with a selection of tense and modal operators, and look at how they
can be used to formalise sentences involving time and modality. The languages
6
These remarks are in no way intended as any criticism of MacFarlane, who remarks on p. 232
that “we are free to use technical terms in whatever way is most useful”, and in fact goes on to
make some very insightful observations, many of which we agree with, pointing out confusions
in certain current debates. One thing is clear. As a quick glance at the articles in two recent
collections on these matters, Preyer and Peter 2005 and 2007, shows, different contributors appear
to want to understand the terms of debate in very different ways.
8
